To the Regency woman what she was wearing on her head was just as - if not more - important as how she had her hair. All proper ladies covered their head when venturing outdoors and any proper married lady always wore something on her head when indoors too.
This is a page from the French fashion magazine Costume Parisienne, from 1811. This was when the Regency fashion era was at it's height, before later fashion trends would start to influence the look. Regency fashion can be said to have lasted from 1793 to circa 1820 and all that time the fashion was accompanied by this form of head-wear.
Hats as we would see them were never worn, hats with brims on all sides of the hat-crown. Instead the thing that everybody wore was the bonnet. The more formal ones, the ones in colour in the picture here, often had the same crown as an "ordinary" hat would have had - made of straw and adorned with ribbons and feathers and plumes. But the brim was made to frame the face and a ribbon was tied under the chin. They were made to protect the wearer from winds, rain and sun.
The other form of bonnet shown here, in white, with less of a sharp shape, are the bonnets supposed to be worn indoors by married women. They were generally of a rather soft material, often adorned with lace and/or ribbons (if only the economy would allow it - if not, a light bonnet would still be worn, but with no trimmings).
To be wearing something on the head was a sure sign of a woman being proper - and the hair was kept to suit the head-wear. This bonnet-fashion would last for the better part of the 19th century, but they would eventually get competition from ordinary hats.